Generalizing Goodstein's theorem and Cichon's independence proof

Abstract

We generalize Goodstein's theorem (Goodstein 1944) and Cichon's independence proof (Cichon 1983) to ${\hat{I}}_1^1-{\mathrm{CA}}_0$ using results from (Wilken 2026). The method is generalizable to stronger notation systems that provide unique terms for ordinals and enjoy Bachmann property.

Major revision and completion of all proofs